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Theorem elwwlks2 41170
 Description: A walk of length 2 between two vertices as length 3 string in a pseudograph. (Contributed by Alexander van der Vekens, 21-Feb-2018.) (Revised by AV, 17-May-2021.)
Hypothesis
Ref Expression
elwwlks2.v 𝑉 = (Vtx‘𝐺)
Assertion
Ref Expression
elwwlks2 (𝐺 ∈ UPGraph → (𝑊 ∈ (2 WWalkSN 𝐺) ↔ ∃𝑎𝑉𝑏𝑉𝑐𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ∃𝑓𝑝(𝑓(1Walks‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))))))
Distinct variable groups:   𝐺,𝑎,𝑏,𝑐,𝑓,𝑝   𝑉,𝑎,𝑏,𝑐,𝑓,𝑝   𝑊,𝑎,𝑏,𝑐,𝑓,𝑝

Proof of Theorem elwwlks2
StepHypRef Expression
1 2nn0 11186 . . 3 2 ∈ ℕ0
2 elwwlks2.v . . . 4 𝑉 = (Vtx‘𝐺)
32wwlksnwwlksnon 41121 . . 3 ((2 ∈ ℕ0𝐺 ∈ UPGraph ) → (𝑊 ∈ (2 WWalkSN 𝐺) ↔ ∃𝑎𝑉𝑐𝑉 𝑊 ∈ (𝑎(2 WWalksNOn 𝐺)𝑐)))
41, 3mpan 702 . 2 (𝐺 ∈ UPGraph → (𝑊 ∈ (2 WWalkSN 𝐺) ↔ ∃𝑎𝑉𝑐𝑉 𝑊 ∈ (𝑎(2 WWalksNOn 𝐺)𝑐)))
52elwwlks2on 41162 . . . 4 ((𝐺 ∈ UPGraph ∧ 𝑎𝑉𝑐𝑉) → (𝑊 ∈ (𝑎(2 WWalksNOn 𝐺)𝑐) ↔ ∃𝑏𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ∃𝑓(𝑓(1Walks‘𝐺)𝑊 ∧ (#‘𝑓) = 2))))
653expb 1258 . . 3 ((𝐺 ∈ UPGraph ∧ (𝑎𝑉𝑐𝑉)) → (𝑊 ∈ (𝑎(2 WWalksNOn 𝐺)𝑐) ↔ ∃𝑏𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ∃𝑓(𝑓(1Walks‘𝐺)𝑊 ∧ (#‘𝑓) = 2))))
762rexbidva 3038 . 2 (𝐺 ∈ UPGraph → (∃𝑎𝑉𝑐𝑉 𝑊 ∈ (𝑎(2 WWalksNOn 𝐺)𝑐) ↔ ∃𝑎𝑉𝑐𝑉𝑏𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ∃𝑓(𝑓(1Walks‘𝐺)𝑊 ∧ (#‘𝑓) = 2))))
8 rexcom 3080 . . . 4 (∃𝑐𝑉𝑏𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ∃𝑓(𝑓(1Walks‘𝐺)𝑊 ∧ (#‘𝑓) = 2)) ↔ ∃𝑏𝑉𝑐𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ∃𝑓(𝑓(1Walks‘𝐺)𝑊 ∧ (#‘𝑓) = 2)))
9 s3cli 13476 . . . . . . . . . 10 ⟨“𝑎𝑏𝑐”⟩ ∈ Word V
109a1i 11 . . . . . . . . 9 ((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) → ⟨“𝑎𝑏𝑐”⟩ ∈ Word V)
11 simplr 788 . . . . . . . . . . . . . . . . 17 (((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → 𝑊 = ⟨“𝑎𝑏𝑐”⟩)
12 simpr 476 . . . . . . . . . . . . . . . . 17 (((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → 𝑝 = ⟨“𝑎𝑏𝑐”⟩)
1311, 12eqtr4d 2647 . . . . . . . . . . . . . . . 16 (((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → 𝑊 = 𝑝)
1413breq2d 4595 . . . . . . . . . . . . . . 15 (((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → (𝑓(1Walks‘𝐺)𝑊𝑓(1Walks‘𝐺)𝑝))
1514biimpd 218 . . . . . . . . . . . . . 14 (((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → (𝑓(1Walks‘𝐺)𝑊𝑓(1Walks‘𝐺)𝑝))
1615com12 32 . . . . . . . . . . . . 13 (𝑓(1Walks‘𝐺)𝑊 → (((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → 𝑓(1Walks‘𝐺)𝑝))
1716adantr 480 . . . . . . . . . . . 12 ((𝑓(1Walks‘𝐺)𝑊 ∧ (#‘𝑓) = 2) → (((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → 𝑓(1Walks‘𝐺)𝑝))
1817impcom 445 . . . . . . . . . . 11 ((((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) ∧ (𝑓(1Walks‘𝐺)𝑊 ∧ (#‘𝑓) = 2)) → 𝑓(1Walks‘𝐺)𝑝)
19 simprr 792 . . . . . . . . . . 11 ((((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) ∧ (𝑓(1Walks‘𝐺)𝑊 ∧ (#‘𝑓) = 2)) → (#‘𝑓) = 2)
20 vex 3176 . . . . . . . . . . . . . . . 16 𝑎 ∈ V
21 s3fv0 13486 . . . . . . . . . . . . . . . . 17 (𝑎 ∈ V → (⟨“𝑎𝑏𝑐”⟩‘0) = 𝑎)
2221eqcomd 2616 . . . . . . . . . . . . . . . 16 (𝑎 ∈ V → 𝑎 = (⟨“𝑎𝑏𝑐”⟩‘0))
2320, 22mp1i 13 . . . . . . . . . . . . . . 15 (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → 𝑎 = (⟨“𝑎𝑏𝑐”⟩‘0))
24 fveq1 6102 . . . . . . . . . . . . . . 15 (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (𝑝‘0) = (⟨“𝑎𝑏𝑐”⟩‘0))
2523, 24eqtr4d 2647 . . . . . . . . . . . . . 14 (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → 𝑎 = (𝑝‘0))
26 vex 3176 . . . . . . . . . . . . . . . 16 𝑏 ∈ V
27 s3fv1 13487 . . . . . . . . . . . . . . . . 17 (𝑏 ∈ V → (⟨“𝑎𝑏𝑐”⟩‘1) = 𝑏)
2827eqcomd 2616 . . . . . . . . . . . . . . . 16 (𝑏 ∈ V → 𝑏 = (⟨“𝑎𝑏𝑐”⟩‘1))
2926, 28mp1i 13 . . . . . . . . . . . . . . 15 (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → 𝑏 = (⟨“𝑎𝑏𝑐”⟩‘1))
30 fveq1 6102 . . . . . . . . . . . . . . 15 (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (𝑝‘1) = (⟨“𝑎𝑏𝑐”⟩‘1))
3129, 30eqtr4d 2647 . . . . . . . . . . . . . 14 (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → 𝑏 = (𝑝‘1))
32 vex 3176 . . . . . . . . . . . . . . . 16 𝑐 ∈ V
33 s3fv2 13488 . . . . . . . . . . . . . . . . 17 (𝑐 ∈ V → (⟨“𝑎𝑏𝑐”⟩‘2) = 𝑐)
3433eqcomd 2616 . . . . . . . . . . . . . . . 16 (𝑐 ∈ V → 𝑐 = (⟨“𝑎𝑏𝑐”⟩‘2))
3532, 34mp1i 13 . . . . . . . . . . . . . . 15 (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → 𝑐 = (⟨“𝑎𝑏𝑐”⟩‘2))
36 fveq1 6102 . . . . . . . . . . . . . . 15 (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (𝑝‘2) = (⟨“𝑎𝑏𝑐”⟩‘2))
3735, 36eqtr4d 2647 . . . . . . . . . . . . . 14 (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → 𝑐 = (𝑝‘2))
3825, 31, 373jca 1235 . . . . . . . . . . . . 13 (𝑝 = ⟨“𝑎𝑏𝑐”⟩ → (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))
3938adantl 481 . . . . . . . . . . . 12 (((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))
4039adantr 480 . . . . . . . . . . 11 ((((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) ∧ (𝑓(1Walks‘𝐺)𝑊 ∧ (#‘𝑓) = 2)) → (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))
4118, 19, 403jca 1235 . . . . . . . . . 10 ((((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) ∧ (𝑓(1Walks‘𝐺)𝑊 ∧ (#‘𝑓) = 2)) → (𝑓(1Walks‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))))
4241ex 449 . . . . . . . . 9 (((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) ∧ 𝑝 = ⟨“𝑎𝑏𝑐”⟩) → ((𝑓(1Walks‘𝐺)𝑊 ∧ (#‘𝑓) = 2) → (𝑓(1Walks‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))))
4310, 42spcimedv 3265 . . . . . . . 8 ((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) → ((𝑓(1Walks‘𝐺)𝑊 ∧ (#‘𝑓) = 2) → ∃𝑝(𝑓(1Walks‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))))
44 1wlklenvp1 40823 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑓(1Walks‘𝐺)𝑝 → (#‘𝑝) = ((#‘𝑓) + 1))
45 simpl 472 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((#‘𝑝) = ((#‘𝑓) + 1) ∧ (#‘𝑓) = 2) → (#‘𝑝) = ((#‘𝑓) + 1))
46 oveq1 6556 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((#‘𝑓) = 2 → ((#‘𝑓) + 1) = (2 + 1))
4746adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((#‘𝑝) = ((#‘𝑓) + 1) ∧ (#‘𝑓) = 2) → ((#‘𝑓) + 1) = (2 + 1))
4845, 47eqtrd 2644 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((#‘𝑝) = ((#‘𝑓) + 1) ∧ (#‘𝑓) = 2) → (#‘𝑝) = (2 + 1))
4948adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑓(1Walks‘𝐺)𝑝 ∧ ((#‘𝑝) = ((#‘𝑓) + 1) ∧ (#‘𝑓) = 2)) → (#‘𝑝) = (2 + 1))
50 2p1e3 11028 . . . . . . . . . . . . . . . . . . . . . . . . 25 (2 + 1) = 3
5149, 50syl6eq 2660 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑓(1Walks‘𝐺)𝑝 ∧ ((#‘𝑝) = ((#‘𝑓) + 1) ∧ (#‘𝑓) = 2)) → (#‘𝑝) = 3)
5251exp32 629 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑓(1Walks‘𝐺)𝑝 → ((#‘𝑝) = ((#‘𝑓) + 1) → ((#‘𝑓) = 2 → (#‘𝑝) = 3)))
5344, 52mpd 15 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓(1Walks‘𝐺)𝑝 → ((#‘𝑓) = 2 → (#‘𝑝) = 3))
5453adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓(1Walks‘𝐺)𝑝 ∧ (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩)) → ((#‘𝑓) = 2 → (#‘𝑝) = 3))
5554imp 444 . . . . . . . . . . . . . . . . . . . 20 (((𝑓(1Walks‘𝐺)𝑝 ∧ (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩)) ∧ (#‘𝑓) = 2) → (#‘𝑝) = 3)
56 eqcom 2617 . . . . . . . . . . . . . . . . . . . . . 22 (𝑎 = (𝑝‘0) ↔ (𝑝‘0) = 𝑎)
5756biimpi 205 . . . . . . . . . . . . . . . . . . . . 21 (𝑎 = (𝑝‘0) → (𝑝‘0) = 𝑎)
58 eqcom 2617 . . . . . . . . . . . . . . . . . . . . . 22 (𝑏 = (𝑝‘1) ↔ (𝑝‘1) = 𝑏)
5958biimpi 205 . . . . . . . . . . . . . . . . . . . . 21 (𝑏 = (𝑝‘1) → (𝑝‘1) = 𝑏)
60 eqcom 2617 . . . . . . . . . . . . . . . . . . . . . 22 (𝑐 = (𝑝‘2) ↔ (𝑝‘2) = 𝑐)
6160biimpi 205 . . . . . . . . . . . . . . . . . . . . 21 (𝑐 = (𝑝‘2) → (𝑝‘2) = 𝑐)
6257, 59, 613anim123i 1240 . . . . . . . . . . . . . . . . . . . 20 ((𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)) → ((𝑝‘0) = 𝑎 ∧ (𝑝‘1) = 𝑏 ∧ (𝑝‘2) = 𝑐))
6355, 62anim12i 588 . . . . . . . . . . . . . . . . . . 19 ((((𝑓(1Walks‘𝐺)𝑝 ∧ (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩)) ∧ (#‘𝑓) = 2) ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) → ((#‘𝑝) = 3 ∧ ((𝑝‘0) = 𝑎 ∧ (𝑝‘1) = 𝑏 ∧ (𝑝‘2) = 𝑐)))
6421wlkpwrd 40822 . . . . . . . . . . . . . . . . . . . . . 22 (𝑓(1Walks‘𝐺)𝑝𝑝 ∈ Word 𝑉)
65 simpr 476 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝐺 ∈ UPGraph ∧ 𝑎𝑉) → 𝑎𝑉)
6665anim1i 590 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) → (𝑎𝑉 ∧ (𝑏𝑉𝑐𝑉)))
67 3anass 1035 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑎𝑉𝑏𝑉𝑐𝑉) ↔ (𝑎𝑉 ∧ (𝑏𝑉𝑐𝑉)))
6866, 67sylibr 223 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) → (𝑎𝑉𝑏𝑉𝑐𝑉))
6968adantr 480 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) → (𝑎𝑉𝑏𝑉𝑐𝑉))
7064, 69anim12i 588 . . . . . . . . . . . . . . . . . . . . 21 ((𝑓(1Walks‘𝐺)𝑝 ∧ (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩)) → (𝑝 ∈ Word 𝑉 ∧ (𝑎𝑉𝑏𝑉𝑐𝑉)))
7170ad2antrr 758 . . . . . . . . . . . . . . . . . . . 20 ((((𝑓(1Walks‘𝐺)𝑝 ∧ (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩)) ∧ (#‘𝑓) = 2) ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) → (𝑝 ∈ Word 𝑉 ∧ (𝑎𝑉𝑏𝑉𝑐𝑉)))
72 eqwrds3 13552 . . . . . . . . . . . . . . . . . . . 20 ((𝑝 ∈ Word 𝑉 ∧ (𝑎𝑉𝑏𝑉𝑐𝑉)) → (𝑝 = ⟨“𝑎𝑏𝑐”⟩ ↔ ((#‘𝑝) = 3 ∧ ((𝑝‘0) = 𝑎 ∧ (𝑝‘1) = 𝑏 ∧ (𝑝‘2) = 𝑐))))
7371, 72syl 17 . . . . . . . . . . . . . . . . . . 19 ((((𝑓(1Walks‘𝐺)𝑝 ∧ (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩)) ∧ (#‘𝑓) = 2) ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) → (𝑝 = ⟨“𝑎𝑏𝑐”⟩ ↔ ((#‘𝑝) = 3 ∧ ((𝑝‘0) = 𝑎 ∧ (𝑝‘1) = 𝑏 ∧ (𝑝‘2) = 𝑐))))
7463, 73mpbird 246 . . . . . . . . . . . . . . . . . 18 ((((𝑓(1Walks‘𝐺)𝑝 ∧ (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩)) ∧ (#‘𝑓) = 2) ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) → 𝑝 = ⟨“𝑎𝑏𝑐”⟩)
75 simprr 792 . . . . . . . . . . . . . . . . . . 19 ((𝑓(1Walks‘𝐺)𝑝 ∧ (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩)) → 𝑊 = ⟨“𝑎𝑏𝑐”⟩)
7675ad2antrr 758 . . . . . . . . . . . . . . . . . 18 ((((𝑓(1Walks‘𝐺)𝑝 ∧ (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩)) ∧ (#‘𝑓) = 2) ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) → 𝑊 = ⟨“𝑎𝑏𝑐”⟩)
7774, 76eqtr4d 2647 . . . . . . . . . . . . . . . . 17 ((((𝑓(1Walks‘𝐺)𝑝 ∧ (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩)) ∧ (#‘𝑓) = 2) ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) → 𝑝 = 𝑊)
7877breq2d 4595 . . . . . . . . . . . . . . . 16 ((((𝑓(1Walks‘𝐺)𝑝 ∧ (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩)) ∧ (#‘𝑓) = 2) ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) → (𝑓(1Walks‘𝐺)𝑝𝑓(1Walks‘𝐺)𝑊))
7978biimpd 218 . . . . . . . . . . . . . . 15 ((((𝑓(1Walks‘𝐺)𝑝 ∧ (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩)) ∧ (#‘𝑓) = 2) ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) → (𝑓(1Walks‘𝐺)𝑝𝑓(1Walks‘𝐺)𝑊))
80 simplr 788 . . . . . . . . . . . . . . 15 ((((𝑓(1Walks‘𝐺)𝑝 ∧ (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩)) ∧ (#‘𝑓) = 2) ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) → (#‘𝑓) = 2)
8179, 80jctird 565 . . . . . . . . . . . . . 14 ((((𝑓(1Walks‘𝐺)𝑝 ∧ (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩)) ∧ (#‘𝑓) = 2) ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) → (𝑓(1Walks‘𝐺)𝑝 → (𝑓(1Walks‘𝐺)𝑊 ∧ (#‘𝑓) = 2)))
8281exp41 636 . . . . . . . . . . . . 13 (𝑓(1Walks‘𝐺)𝑝 → ((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) → ((#‘𝑓) = 2 → ((𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)) → (𝑓(1Walks‘𝐺)𝑝 → (𝑓(1Walks‘𝐺)𝑊 ∧ (#‘𝑓) = 2))))))
8382com25 97 . . . . . . . . . . . 12 (𝑓(1Walks‘𝐺)𝑝 → (𝑓(1Walks‘𝐺)𝑝 → ((#‘𝑓) = 2 → ((𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)) → ((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) → (𝑓(1Walks‘𝐺)𝑊 ∧ (#‘𝑓) = 2))))))
8483pm2.43i 50 . . . . . . . . . . 11 (𝑓(1Walks‘𝐺)𝑝 → ((#‘𝑓) = 2 → ((𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)) → ((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) → (𝑓(1Walks‘𝐺)𝑊 ∧ (#‘𝑓) = 2)))))
85843imp 1249 . . . . . . . . . 10 ((𝑓(1Walks‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) → ((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) → (𝑓(1Walks‘𝐺)𝑊 ∧ (#‘𝑓) = 2)))
8685com12 32 . . . . . . . . 9 ((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) → ((𝑓(1Walks‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) → (𝑓(1Walks‘𝐺)𝑊 ∧ (#‘𝑓) = 2)))
8786exlimdv 1848 . . . . . . . 8 ((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) → (∃𝑝(𝑓(1Walks‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))) → (𝑓(1Walks‘𝐺)𝑊 ∧ (#‘𝑓) = 2)))
8843, 87impbid 201 . . . . . . 7 ((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) → ((𝑓(1Walks‘𝐺)𝑊 ∧ (#‘𝑓) = 2) ↔ ∃𝑝(𝑓(1Walks‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))))
8988exbidv 1837 . . . . . 6 ((((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) ∧ 𝑊 = ⟨“𝑎𝑏𝑐”⟩) → (∃𝑓(𝑓(1Walks‘𝐺)𝑊 ∧ (#‘𝑓) = 2) ↔ ∃𝑓𝑝(𝑓(1Walks‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2)))))
9089pm5.32da 671 . . . . 5 (((𝐺 ∈ UPGraph ∧ 𝑎𝑉) ∧ (𝑏𝑉𝑐𝑉)) → ((𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ∃𝑓(𝑓(1Walks‘𝐺)𝑊 ∧ (#‘𝑓) = 2)) ↔ (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ∃𝑓𝑝(𝑓(1Walks‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))))))
91902rexbidva 3038 . . . 4 ((𝐺 ∈ UPGraph ∧ 𝑎𝑉) → (∃𝑏𝑉𝑐𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ∃𝑓(𝑓(1Walks‘𝐺)𝑊 ∧ (#‘𝑓) = 2)) ↔ ∃𝑏𝑉𝑐𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ∃𝑓𝑝(𝑓(1Walks‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))))))
928, 91syl5bb 271 . . 3 ((𝐺 ∈ UPGraph ∧ 𝑎𝑉) → (∃𝑐𝑉𝑏𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ∃𝑓(𝑓(1Walks‘𝐺)𝑊 ∧ (#‘𝑓) = 2)) ↔ ∃𝑏𝑉𝑐𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ∃𝑓𝑝(𝑓(1Walks‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))))))
9392rexbidva 3031 . 2 (𝐺 ∈ UPGraph → (∃𝑎𝑉𝑐𝑉𝑏𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ∃𝑓(𝑓(1Walks‘𝐺)𝑊 ∧ (#‘𝑓) = 2)) ↔ ∃𝑎𝑉𝑏𝑉𝑐𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ∃𝑓𝑝(𝑓(1Walks‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))))))
944, 7, 933bitrd 293 1 (𝐺 ∈ UPGraph → (𝑊 ∈ (2 WWalkSN 𝐺) ↔ ∃𝑎𝑉𝑏𝑉𝑐𝑉 (𝑊 = ⟨“𝑎𝑏𝑐”⟩ ∧ ∃𝑓𝑝(𝑓(1Walks‘𝐺)𝑝 ∧ (#‘𝑓) = 2 ∧ (𝑎 = (𝑝‘0) ∧ 𝑏 = (𝑝‘1) ∧ 𝑐 = (𝑝‘2))))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∃wrex 2897  Vcvv 3173   class class class wbr 4583  ‘cfv 5804  (class class class)co 6549  0cc0 9815  1c1 9816   + caddc 9818  2c2 10947  3c3 10948  ℕ0cn0 11169  #chash 12979  Word cword 13146  ⟨“cs3 13438  Vtxcvtx 25673   UPGraph cupgr 25747  1Walksc1wlks 40796   WWalkSN cwwlksn 41029   WWalksNOn cwwlksnon 41030 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-ac2 9168  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-ifp 1007  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-1o 7447  df-2o 7448  df-oadd 7451  df-er 7629  df-map 7746  df-pm 7747  df-en 7842  df-dom 7843  df-sdom 7844  df-fin 7845  df-card 8648  df-ac 8822  df-cda 8873  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-n0 11170  df-z 11255  df-uz 11564  df-fz 12198  df-fzo 12335  df-hash 12980  df-word 13154  df-concat 13156  df-s1 13157  df-s2 13444  df-s3 13445  df-uhgr 25724  df-upgr 25749  df-edga 25793  df-1wlks 40800  df-wlks 40801  df-wwlks 41033  df-wwlksn 41034  df-wwlksnon 41035 This theorem is referenced by: (None)
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